Kahle [1]} developed an important instance of this object in work on certain probability spaces of simplicial complexes. Delgado-Friedrichs et al. considered an analogous construction for cubical complexes in [2]}, which names the elements of \(\mathrm {Par}(D)\)  close pairs. Independently, Henselman-Petrusek defined \(\mathrm {Par}(D)\)  for arbitrary boundary matrices under the name Pareto frontier in [3]}, and later termed the elements of this set minimal pairs [4]}. Bauer generalized the construction of Kahle to arbitrary filtered simplicial complexes for use in PH computations, in work that ultimately appeared in [5]}. Following the appearance of that work, Lampret independently developed a more general construction called a steepness matching; unlike those which preceded it, the steepness matching is suitable for boundary matrices with coefficients in an arbitrary unital ring (by contrast, the predecessors restricted to field coefficients).
